HOLT-WINTERS AND NEURAL-NETWORK METHODS FOR ...

HOLT-WINTERS AND NEURAL-NETWORK METHODS FOR MEDIUM-TERM SALES FORECASTING Apostolos Kotsialos, Markos Papageorgiou, and Antonios Poulimenos Dynamic Systems and Simulation Laboratory, Technical University of Crete, 73100 Chania, Greece Phone: +30-28210-37289 Fax: +30-28210- 37584 E-mail: [email protected]

Abstract: The problem of medium to long term sales forecasting raises a number of requirements that must be suitably addressed in the design of the employed forecasting methods. These include long forecasting horizons (up to 52 periods ahead), a high number of quantities to be forecasted, which limits the possibility of human intervention, as well as frequent introduction of new articles (for which no past sales are available for parameter calibration) and withdrawal of running articles. The problem has been tackled by use of a modified Holt-Winters method as well as Feedforward Multilayer Neural Networks (FMNN) applied to sales data from two German companies. Copyright © 2005 IFAC Keywords: Forecasts, feedforward neural networks, time-series analysis

1. INTRODUCTION Sales forecasting over a sufficiently long future time horizon is an important prerequisite for efficient production planning and a solid basis for company policy decisions. A number of efficient forecasting algorithms with different levels of complexity have been developed and tested in the past. Most method assessments and comparisons, however, address the problem of short-term or even one-step-ahead forecasting (see, e.g. Makridakis et al., 1982; Gaynor and Kirkpatrick, 1994; Lachtermacher and Fuller, 1995). A hybrid approach combining long-term with shortterm forecasts employing the Holt-Winters method was investigated by Rajopadhye et al. (1999). The forecasting methods to be employed should be able to operate largely autonomously, because human supervision and intervention is hardly possible in case of thousands of article sales to be forecasted. The well known Holt-Winters method (Chatfield, 1978) is suitably modified for long-term forecasting, and various Feedforward Multilayer Neural Network (FMNN) approaches are also proposed for the same problem. Both groups of methods as well as combinations thereof are applied to various kinds of articles, group sales, and total sales from two Ger-

man companies (a total of 195 time-series are used) to assess and compare their forecasting performance and their suitability in view of the above requirements. FMNN-based prediction has recently gained remarkable popularity due to the possibility to describe complex nonlinear interrelationships within a relatively convenient black-box approach (Chakraborty et al., 1992; Yao, 1999). FMNN methods have been increasingly applied to prediction problems during the last decade (Chakraborty et al., 1992; Cottrell, et al., 1995; Lachtermacher and Fuller, 1995; Bunn, 2000). Their basic advantage is that they may capture the unknown nonlinear structure of the process to be modelled. Their basic disadvantages are the high number of parameters to be calibrated and the blackbox approach that renders the plausible interpretation of the modeling structure very difficult and excludes the possibility of ad-hoc parameter choice. 2. AVAILABLE DATA A total of 195 time-series is used to test and compare the various versions of Holt-Winters and FMNNbased predictors. All time-series represent sales quantities from two industrial firms, namely the toy

Table 1 Average of seasonality correlation factor Group D1 D2 D3 D4 rs 0.29 0.36 0.06 0.016 max rs 0.53 0.45 0.18 0.08 -0.03 0.26 -0.07 0.0 min rs Group D5 D6 D7 D8 rs 0.25 0.48 -0.19 0.02 max rs 0.56 0.48 0.27 0.18 min rs -0.13 0.48 -0.51 -0.08 producer sigikid (sk) and a wire-harness production factory for trucks owned by DaimlerChrysler (DC). All available time-series extend over the years 19971999 on a weekly base. For sk-data we have, for each time-series, 52 values for 1997, 52 values for 1998, and 40 values for 1999, while DC-data cover, for each time-series, 52 values for each of the three years. For a better analysis of results, the 195 timeseries are assigned to the following 8 data groups: D1: 60 time-series of sales numbers for 60 ordinary sk-articles with various average sales numbers . D2: 4 time-series of sales values (in german DM) for 4 sk groups of articles with various average values. D3: 10 time-series of sales numbers for sk-articles which were withdrawn from production during the period 1997-1999. D4: 5 time-series of sales numbers for sk-articles which were first introduced within the period 19971999. D5: 100 time-series as in D1 but for DC-articles. D6: 1 time-series of sales numbers for a DC-group of articles. D7: 10 time-series as in D3 but for DC-articles. D8: 5 time-series as in D4 but for DC-articles. The available data display some kind of more or less pronounced periodicity of the time-series over the time span of the year (i.e. over 52 weeks). This periodicity reflects the yearly seasonality of sales that may depend on annual fairs, end-of-year or summer holidays, etc. Assume a weekly time-series Yt, t=1,…,52, over a year, with average value L=

1 52

52

∑ Yt .

(1)

t =1

The seasonality factors St, t=1,…,52, for the weeks of that year are then defined as St = Yt/L. For each time-series from D1-D8, the corresponding seasonality factors S t1997 and S t1998 , t=1,…,52, for the years 1997 and 1998, respectively, may be calculated. To avoid later difficulties, St = 0.1 is set whenever a value Yt≤0 is encountered in the data (negative values correspond to very limited article returns). In order to calculate a practical index for the strength of seasonality, a simple correlation formula is applied to the 1997 and 1998 seasonality factors for each timeseries as following

∑ (St1997 − 1)(St1998 − 1) 52

rs =

t =1

∑ (St1997 − 1) ∑ (St1998 − 1) 52

t =1

2

52

t =1

2

(2)

where rs is the seasonality correlation factor. The higher rs is for a particular time-series, the stronger the seasonality pattern. A value of close to zero indicates no seasonality while rs